To be ‘fancy free’ is to be without any ties or commitments.
‘Fancy free’ is one of the numerous words and expressions coined by William Shakespeare and is first found in A Midsummer Nights Dream, 1598:

OBERON:
But I might see young Cupid’s fiery shaft
Quench’d in the chaste beams of the watery moon,
And the imperial votaress passed on,
In maiden meditation, fancy-free.